30 research outputs found
Nonic 3-adic Fields
We compute all nonic extensions of Q3 and find that there are 795 of them up to isomorphism. We describe how to compute the associated Galois group of such a field, and also the slopes measuring wild ramification. We present summarizing tables and a sample application to number fields
A Database of Local Fields
We describe our online database of finite extensions of Qp, and how it can be used to facilitate local analysis of number fields
The tame-wild principle for discriminant relations for number fields
Consider tuples of separable algebras over a common local or global number
field, related to each other by specified resolvent constructions. Under the
assumption that all ramification is tame, simple group-theoretic calculations
give best possible divisibility relations among the discriminants. We show that
for many resolvent constructions, these divisibility relations continue to hold
even in the presence of wild ramification.Comment: 31 pages, 11 figures. Version 2 fixes a normalization error: |G| is
corrected to n in Section 7.5. Version 3 fixes an off-by-one error in Section
6.
Torsion Subgroups of Rational Elliptic Curves Over Odd Degree Galois Fields
The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with Mazur\u27s classification of the possible torsion subgroups E(Q)_tors for rational elliptic curves, and there has since been an explosion of classifications.
Inspired by work of Chou, González Jiménez, Lozano-Robledo, and Najman, the purpose of this work is to classify the set Φ_Q^{Gal}(9), i.e. the set of possible torsion subgroups for rational elliptic curves over nonic Galois fields. We not only completely determine the set Φ_Q^{Gal}(9), but we also determine the possible torsion subgroups based on the isomorphism type of Gal(K/Q). We then determine the possibilities for the growth of torsion from E(Q)_tors to E(K)_tors, i.e. what the possibilities are for E(K)_tors ⊇ E(Q)_tors given a fixed torsion subgroup E(Q)_tors. Extending the techniques used in the classification of Φ_Q^{Gal}(9), we then determine the possible structures over all odd degree Galois fields. Finally, we explicitly determine the sets Φ_Q^{Gal}(d) for all odd d based on the prime factorization for d while proving a number of other related results
On the index divisors and monogenity of certain nonic number fields
In this paper, for any nonic number field generated by a root of
a monic irreducible trinomial and for every
rational prime , we characterize when divides the index of . We also
describe the prime power decomposition of the index . In such a way we
give a partial answer of Problem of Narkiewicz (\cite{Nar}) for this
family of number fields. In particular if , then is not
mongenic. We illustrate our results by some computational examples.Comment: arXiv admin note: text overlap with arXiv:2112.0113
On relative pure cyclic fields with power integral bases
summary:Let be an extension of a number field , where satisfies the monic irreducible polynomial of prime degree belonging to ( is the ring of integers of ). The purpose of this paper is to study the monogenity of over by a simple and practical version of Dedekind's criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field with a pure cubic subfield, which is not necessarily a composite extension of two cubic subfields. We obtain a slightly simpler computation of the discriminant